ECO 305 — FALL 2003 — September 25

INDIRECT UTILITY FUNCTION

U∗(Px, Py,M) = max {U(x, y) |Px x+ Py y ≤M }= U(x∗, y∗)= U(Dx(Px, Py,M), Dy(Px, Py,M) )

PROPERTIES OF U∗:(1) No money illusion — Homogeneous degree zero:

U∗(k Px, k Py, kM) = U∗(Px, Py,M)

(2) As money income changes:

∂U∗

∂M=

∂U

∂x

¯̄̄̄¯∗

∂x∗

∂M+

∂U

∂y

¯̄̄̄¯∗

∂y∗

∂M

= λ

"Px

∂x∗

∂M+ Py

∂y∗

∂M

#

= λ∂M

∂M= λ

(3) As price changes:

∂U∗

∂Px=

∂U

∂x

¯̄̄̄¯∗

∂x∗

∂Px+

∂U

∂y

¯̄̄̄¯∗

∂y∗

∂Px

1

= λ

"Px

∂x∗

∂Px+ Py

∂y∗

∂Px

#= −λ x∗ (just like M ↓ by x∗)

(Last step: differentiate adding-up identity w.r.t. Px:

Px x∗ + Py y∗ =M

x∗ + Px∂x∗

∂Px+ Py

∂y∗

∂Px= 0 )

Divide price- and income-change equations :

Roy’s Identity: x∗ = − ∂U∗/∂Px∂U∗/∂M

(4) Contours of U∗ in (Px, Py) space with M fixed:

(Like theater with stage at NE corner)

2

EXPENDITURE FUNCTION

Solve the indirect utility function for income:

u = U∗(Px, Py,M) ⇐⇒ M =M∗(Px, Py, u)

M∗(Px, Py, u) = min {Px x+ Py y |U(x, y) ≥ u }“Dual” or mirror image of utility maximization problem.Economics — income compensation for price changesOptimum quantities — Compensated or Hicksian demands

x∗ = DHx (Px, Py, u) , y∗ = DH

y (Px, Py, u)

PROPERTIES OF M∗:(1) Homogeneous degree 1 in (Px, Py) holding u fixed:

M∗(k Px, k Py, u) = k M∗(Px, Py, u)

(2) Hotelling’s or Shepherd’s Lemma —Compensated demands partial derivatives w.r.t. prices:

DHx (Px, Py, u) = ∂M∗/∂Px , DH

y (Px, Py, u) = ∂M∗/∂Py

Proof: M∗ = Px DHx + Py D

Hy , u = U(D

Hx , D

Hy ). So

∂M∗/∂Px = DHx + Px ∂DH

x /∂Px + Py ∂DHy /∂Px

0 = Ux ∂DHx /∂Px + Uy ∂DH

y /∂Px

= λ [Px ∂DHx /∂Px + Py ∂DH

y /∂Px ]

3

(3) “Weakly” concave in (Px, Py) holding u fixed.Cobb-Douglas example: (Px)

1/3 (Py)2/3

PROPERTIES OF HICKSIAN DEMAND FUNCTIONS:

(1) Own substitution effect negative:

∂x

∂Px

¯̄̄̄¯u=const

=∂DH

x

∂Px=

∂2M∗

∂P 2x≤ 0

(2) Symmetry of cross-price effects:

∂DHx

∂Py=

∂2M∗

∂Px∂Py=

∂DHy

∂Px

(Net) substitutes if > 0, complements if < 0General concept : Comparative statics

4

COBB-DOUGLAS EXAMPLE

(Direct) UTILITY FUNCTION:

U(x, y) = α ln(x) + β ln(y), α+ β = 1

x∗ = αM/Px, y∗ = βM/Py

INDIRECT UTILITY FUNCTION

U∗(Px, Py,M) = α [ln(α) + ln(M)− ln(Px) ]+β [ln(β) + ln(M)− ln(Py) ]

= junk+ ln(M)− α ln(Px)− β ln(Py)

Roy’s Identity:

− ∂U∗/∂Px∂U∗/∂M

= − −α/Px1/M

=αM

Px= x∗

EXPENDITURE FUNCTION

M∗ =M∗(Px, Py, u) = eu (Px)α (Py)β

Hicksian demand functions

xH = α eu (Px)α−1 (Py)β, yH = β eu (Px)

α (Py)β−1

5

SLUTSKY EQUATION

Link between Marshallian and Hicksian demandsEqual if u = U∗(Px, Py,M), M =M∗(Px, Py, u).

For good i where i may be either x or y,

DHi (Px, Py, u) = D

Mi (Px, Py,M

∗(Px, Py, u) )

Now let Pj change, where j may be x or y

∂DHi

∂Pj=

∂DMi

∂Pj+

∂DMi

∂M

∂M∗

∂Pj

=∂DM

i

∂Pj+

∂DMi

∂MDHj

=∂DM

i

∂Pj+

∂DMi

∂MDMj

For example

∂x

∂Py

¯̄̄̄¯u=const

=∂x

∂Py

¯̄̄̄¯M=const

+ y∂x

∂M

Price derivative of compensated demand =Price derivative of uncompensated demand+ Income effect of compensation.

If i = j, LHS is negative. Then Giffen implies Inferior

6